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We prove the hard Lefschetz duality for locally conformally almost Kähler manifolds. This is a generalization of that for almost Kähler manifolds studied by Cirici and Wilson. We generalize the Kähler identities to prove the duality. Based on the result, we introduce the hard Lefschetz condition for locally conformally symplectic manifolds. As examples, we give solvmanifolds which do not satisfy the hard Lefschetz condition.

In this paper we study generalized Kenmotsu manifolds (shortly, a GK-manifold) that satisfy the axiom of Φ-holomorphic (2r+1)-planes. After the preliminaries we give the definition of generalized Kenmotsu manifolds and the full structural equation group. Next, we define Φ- holomorphic generalized Kenmotsu manifolds and Φ-paracontact generalized Kenmotsu manifold give a local characteristic of this subclasses. The Φ-holomorphic generalized Kenmotsu manifold coincides with the class of almost contact metric manifolds obtained from closely cosymplectic manifolds by a canonical concircular transformation of nearly cosymplectic structure. A Φ- paracontact generalized Kenmotsu manifold is a special generalized Kenmotsu manifold of the second kind. An analytical expression is obtained for the tensor of Ф-holomorphic sectional curvature of generalized Kenmotsu manifolds of the pointwise constant Φ-holomorphic sectional curvature. Then we study the axiom of Φ-holomorphic (2r+1)-planes for generalized Kenmotsu manifolds and propose a complete classification of simply connected generalized Kenmotsu manifolds satisfying the axiom of Φ-holomorphic (2r+1)-planes. The main results are as follows. A simply connected GK-manifold of pointwise constant Φ-holomorphic sectional curvature satisfying the axiom of Φ-holomorphic (2r+1)-planes is a Kenmotsu manifold. A GK-manifold satisfies the axiom of Φ-holomorphic (2r+1)-planes if and only if it is canonically concircular to one of the following manifolds: (1) CPn×R; (2) Cn×R; and (3) CHn×R having the canonical cosymplectic structure.

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